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The problem presented is a classic example of a stochastic process known as the Gambler's Ruin problem. In this scenario, a gambler starts with a certain amount of money and continues to gamble until they either lose all their money or reach a target amount. The key here is to determine the probability of the gambler reaching the target amount before going broke.
Understanding the Random Walk:
Expected Value and Probability:
Setting Up the Equation:
Let r
be the probability of reaching $100.
The probability of reaching $0 is 1 - r
.
The expected value equation can be set up as follows:
E[X]=100×r+0×(1−r)=100r
Since we know that the expected value is $30, we substitute:
30=100r
Solving for r
gives us:
r=10030=0.3
Conclusion:
Markov Chain Perspective:
Symmetry and Fairness:
Practical Implications:
This explanation provides a comprehensive understanding of the problem and its solution, integrating concepts from probability theory and stochastic processes.